Corollary

For every projector e of E, E is the direct sum of the image $M = e(E)$ and the kernel $N = e^{-1}(0)$ of e; for all $x = x_1 + x_2 \in E$ with $x_1 \in M$ and $x_2 \in N$, $x_1 = e(x)$; 1 − e is a projector of E of image N and kernel M.

$(1 - e)^2 = 1 - 2e + e^2 = 1 - e$ in $\operatorname{End}(E)$ and hence $1 - e$ is a projector; as also $e(1 - e) = (1 - e)e = e - e^2 = 0$, E is the direct sum of the images $M$ and $N$ of e and $1 - e$ by Proposition 12. Finally, for all $x \in E$, the relation $x \in M$ is equivalent to $x = e(x)$; for $x = e(x)$ implies by definition $x \in M$ and, conversely, if $x = e(x')$ with $x' \in E$, then $e(x) = e^2(x') = e(x') = x;$

this shows therefore that M is the kernel of $1 - e$ and, exchanging the roles of $e$ and $1 - e$, it is similarly seen that N is the kernel of $e$.